Least upper bound
An upper bound of is the least upper bound if for every upper bound of .
Definition 5.4 (Least Upper Bound)
An upper bound of a set is a least upper bound if for any upper bound of , . We write for the least upper bound of .
Supremum versus maximum
If has a greatest element then , but a least upper bound need not belong to : for we have and , and likewise although every element is smaller than .
Related
Stated in
- Definition 5.4 (Least Upper Bound)§5.1 Construction of